Linear Algebra Answers

2x+5y+3z=2

x+2y+2z=4

x+y+4z=11. Use the elementary row reduction to solve for the linear system.

Let S = {w1,w2,...,wk} be a basis for the vector space V. Prove that every vector in V can

be expressed as a linear combination of vectors w1,w2,...,wk

in exactly one .

Let W be the set of all 3×3 real diagonal matrices. Prove that W is a subspace of M33.

1.9.given A "\\begin{bmatrix}\n 1 & 1& -1 \\\\\n 2 & 2 & 3 \\\\\n 4 & 0 &0 \\\\\n \n\\end{bmatrix}" B="\\begin{bmatrix}\n 1 & 1 & -1 \\\\\n 2 & 2 & 3 \\\\\n 4 & 0 & 0 \\\\\n\n\\end{bmatrix}"

1.9.1. find -A ^-1 +3B^T

1.9.2. Find B^-1+(A^T+A^-1)

1.use Cramer's rule to solve the equation below (10 marks)

y-z =2

3x+2y+z=4

5x+4y=1

Use Descartes’ Rule of Signs to determine the possible number of positive, negative and

imaginary zeros of P (x).

(i) P (x) = 2x^3+x^2-3x+6.

Use Descartes’ Rule of Signs to determine the possible number of positive, negative and

imaginary zeros of P (x).

(i) P (x) = 2x^5+x^4+x^3-4x^2-x-3.

A, B, and C are on a betting game. B loses P350 of his money to A. As a result, A now has twice as much as what is left with B. Then, C loses P700 to B. As a consequence, C now has only one-third as much money as B would then have. If A loses P210 to C, C will have as much money as A would have left. How much did each have at the start?

A weight A lbs. on one side of a beam balances a weight of 40 lbs. placed 6ft from the fulcrum on the other side. If the unknown weight is moved 3ft nearer the fulcrum, it balances a weight of 20lbs placed ft from 7 1/2 the fulcrum. Find the unknown weight and its distance from the fulcrum in the first instance. (Neglect the weight of the beam.)

If the numerator of a fraction is doubled and the denominator is increased by 7, the value becomes 2/3; but if the denominator is doubled and the numerator is increased by 2, the value becomes 3/5. What is the original fraction?